onstruct a truth table for the statement.
left parenthesis tilde p logical or tilde r right parenthesis left right arrow left parenthesis tilde r logical or q right parenthesis
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Part 1
Construct a truth table for the statement. Fill in the blanks below.
p
q
r
​(tilde p
logical or
tilde r​)
left right arrow
​(tilde r
logical or
q​)
T
T
T
▼ 
T
F
▼ 
F
T
Upper F
▼ 
T
F
Upper F


To construct a truth table for the given statement \(\left(\sim p \lor \sim r \right) \leftrightarrow \left(\sim r \lor q \right)\), we will create a table with all possible combinations of truth values for \(p\), \(q\), and \(r\). Then, we will compute the values of the sub-expressions step by step, which includes negating variables and evaluating the logical disjunction (OR, \(\lor\)) and biconditional (\(\leftrightarrow\)) operations.

### Key Symbols:
- \(\sim p\): Negation of \(p\)
- \(\lor\): Logical OR
- \(\leftrightarrow\): Biconditional ("if and only if")

### Step-by-step process:
1. **Column 1-3**: Assign all possible truth values for \(p\), \(q\), and \(r\).
2. **Column 4**: Compute \(\sim p\) and \(\sim r\), then find the value of \(\sim p \lor \sim r\).
3. **Column 5**: Compute \(\sim r\), then find the value of \(\sim r \lor q\).
4. **Column 6**: Evaluate the biconditional \(\left(\sim p \lor \sim r\right) \leftrightarrow \left(\sim r \lor q\right)\).

Let's build the truth table step-by-step.

| \(p\)  | \(q\)  | \(r\)  | \(\sim p \lor \sim r\) | \(\sim r \lor q\) | \(\left(\sim p \lor \sim r\right) \leftrightarrow \left(\sim r \lor q\right)\) |
|-------|-------|-------|-----------------------|------------------|--------------------------------------------------------------------------|
| **T** | **T** | **T** | **F**                 | **T**            | **F**                                                                    |
| **T** | **T** | **F** | **T**                 | **T**            | **T**                                                                    |
| **T** | **F** | **T** | **F**                 | **F**            | **T**                                                                    |
| **T** | **F** | **F** | **T**                 | **F**            | **F**                                                                    |
| **F** | **T** | **T** | **T**                 | **T**            | **T**                                                                    |
| **F** | **T** | **F** | **T**                 | **T**            | **T**                                                                    |
| **F** | **F** | **T** | **T**                 | **F**            | **F**                                                                    |
| **F** | **F** | **F** | **T**                 | **F**            | **F**                                                                    |

### Explanation:
- **\(p\), \(q\), and \(r\)**: The first three columns list all possible combinations of truth values for \(p\), \(q\), and \(r\) (T = true, F = false).
- **\(\sim p \lor \sim r\)**: This column shows the result of the disjunction (OR) between the negation of \(p\) and the negation of \(r\).
- **\(\sim r \lor q\)**: This column shows the result of the disjunction (OR) between the negation of \(r\) and \(q\).
- **\(\left(\sim p \lor \sim r\right) \leftrightarrow \left(\sim r \lor q\right)\)**: The biconditional checks if the two sub-expressions have the same truth value. If they do, the result is T (True); otherwise, it is F (False).

Let me know if you'd like more details or explanations!

### 5 Related Questions:
1. What does the symbol \(\leftrightarrow\) mean in logical expressions?
2. How do you compute the negation (\(\sim\)) of a truth value?
3. What is the difference between logical OR (\(\lor\)) and logical AND (\(\land\))?
4. How can you verify if two logical expressions are logically equivalent?
5. How does the truth table change if we modify one of the sub-expressions?

### Tip:
Always break down complex logical expressions into smaller components when constructing truth tables!

Construct a truth table for the statement (tilde p logical or tilde r) left right arrow (tilde r logical or q).

Learn how to construct a truth table for the logical expression (~p ∨ ~r) ↔ (~r ∨ q). This problem involves negation, logical OR, and biconditional operations, providing a step-by-step guide suitable for high school students in Grades 9-12.

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